few_shots: Example:
Problem/activity:
Stage 1: Money
Required Preparation
Materials to Copy
Blackline Masters
Would You Rather Stage 1 Spinner
Would You Rather Stage 1 Recording Sheet

Narrative
The first partner spins to get a group of coins. They write a question that compares the amount they spun to a different group of coins that they make up.


Mathematical concept/skill options: 
A. Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.
B. Represent and interpret data.
C. Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.
D. Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.
E. Describe and compare measurable attributes.
F. Geometric measurement: understand concepts of angle and measure angles.
G. Convert like measurement units within a given measurement system.
H. Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
I. Measure and estimate lengths in standard units.
J. Classify objects and count the number of objects in each category.
K. Tell and write time.
L. Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
M. Measure lengths indirectly and by iterating length units.
N. Work with time and money.
O. Relate addition and subtraction to length.


Thought: In this problem, students work with money in the form of coins.
Answer: N

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: im_center_000269

2
options: Solve real-world and mathematical problems involving area, surface area, and volume.

Prove geometric theorems.

Draw, construct, and describe geometrical figures and describe the relationships between them.

Graph points on the coordinate plane to solve real-world and mathematical problems.

Translate between the geometric description and the equation for a conic section.

Understand and apply the Pythagorean Theorem.

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

Classify two-dimensional figures into categories based on their properties.

Understand and apply theorems about circles.

Prove theorems involving similarity.

Define trigonometric ratios and solve problems involving right triangles.

Understand similarity in terms of similarity transformations.

Make geometric constructions.

Visualize relationships between two-dimensional and three-dimensional objects.

Explain volume formulas and use them to solve problems.

Find arc lengths and areas of sectors of circles.

Apply trigonometry to general triangles.

Reason with shapes and their attributes.

Apply geometric concepts in modeling situations.

Analyze, compare, create, and compose shapes.

Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).

Understand congruence and similarity using physical models, transparencies, or geometry software.

Experiment with transformations in the plane.

Use coordinates to prove simple geometric theorems algebraically.

Understand congruence in terms of rigid motions.

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
problem_activity: Optional activity
In this activity, the dots are distributed uniformly in the first square but not in the second square:
[IMG]
However, these dots are drawn so it is not too hard to see that if they were redistributed, each square inch would have 8 dots:
[IMG]
The fact that we have 8 dots per square inch means that if we distributed the dots uniformly throughout the square in the right way, we would see 8 dots in each square inch. This prepares students to be able to interpret the constant of proportionality in the next two activities when working with actual houses per square mile or people per square kilometer. When we speak of “500 houses per square kilometer,” we can think of this as, “If we took all of the houses in the region and spread them out uniformly, then we would see 500 houses in every square kilometer.”
Digital
Launch
Display the image of the two squares with dots. Ask students to describe what is the same and what is different about these squares.
Give students 2–3 minutes of quiet work time, followed by 4–5 minutes of partner work time and whole-class discussion.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, pause to check for understanding after the first 2–3 minutes of work time.
Supports accessibility for: Organization; Attention
Writing, Conversing: MLR5 Co-craft Questions.
Display only the image of the four dot arrays without revealing any of the questions that follow. Give students 1–2 minutes to write a list of possible mathematical questions they could ask about the arrays. Invite students to share their questions with a partner, and then select 2–3 students to share their questions with the whole class. Highlight any questions that refer to how dots are “distributed,” even if students do not use that particular phrase. Finally, reveal the whole problem with text so that students may begin addressing the questions. This helps amplify language related to the distribution of dots.
Design Principle(s): Cultivate conversation; Maximize meta-awareness
Student Facing
The figure shows two arrays, each enclosed by a square that is 2 inches wide.
[IMG]
Let
\(a\)
be the area of the square and
\(d\)
be the number of dots enclosed by the square. For each square, plot a point that represents its values of
\(a\)
and
\(d\)
. Start by choosing an appropriate scale for each axis, using the sliders to adjust the intervals.
Draw lines from
\((0, 0)\)
to each point. For each line, write an equation that represents the proportional relationship.
What is the constant of proportionality for each relationship? What do the constants of proportionality tell us about the dots and squares?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Display the image of the two squares with dots. Ask students to describe what is the same and what is different about these squares.
Give students 2–3 minutes of quiet work time, followed by 4–5 minutes of partner work time and whole-class discussion.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, pause to check for understanding after the first 2–3 minutes of work time.
Supports accessibility for: Organization; Attention
Writing, Conversing: MLR5 Co-craft Questions.
Display only the image of the four dot arrays without revealing any of the questions that follow. Give students 1–2 minutes to write a list of possible mathematical questions they could ask about the arrays. Invite students to share their questions with a partner, and then select 2–3 students to share their questions with the whole class. Highlight any questions that refer to how dots are “distributed,” even if students do not use that particular phrase. Finally, reveal the whole problem with text so that students may begin addressing the questions. This helps amplify language related to the distribution of dots.
Design Principle(s): Cultivate conversation; Maximize meta-awareness
Student Facing
The figure shows two arrays, each enclosed by a square that is 2 inches wide.
[IMG]
Let
\(a\)
be the area of the square and
\(d\)
be the number of dots enclosed by the square. For each square, plot a point that represents its values of
\(a\)
and
\(d\)
.
[IMG]
Draw lines from
\((0, 0)\)
to each point. For each line, write an equation that represents the proportional relationship.
What is the constant of proportionality for each relationship? What do the constants of proportionality tell us about the dots and squares?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
The goal of this discussion is for students to make sense of the constant of proportionality in the case where the dots are not uniformly distributed. Invite students to share their interpretations of the constants of proportionality.
Consider asking questions like:
“Why is one of the constants of proportionality larger than the other? How can we see this in the picture of the squares?” (There are more dots in the same area.)
“What are the units of the constants of proportionality?” (The number of dots per square inch).
Students will have more opportunities to think about this when working on the activities that follow.


correct_option_index: 6